The verbless clause she speaks, מִי־אַ֣תְּ בִּתִּ֑י, is variously translated into English in other ways as well:
KJV - Who art thou, my daughter? [The same idea as the JPS quoted in the question]
NKJV - Is that you, my daughter?
ESV - How did you fare, my daughter?
NASB/NIV - How did it go, my daughter?
The only consistent part is the "my daughter" translating the בִּתִּ֑י.
Most translations also correctly have the "you" for the feminine singular אַ֣תְּ. The NASB/NIV "it" is odd, since the word is a 2nd person pronoun, not a 3rd person pronoun. However, when viewed with the ESV rendering, one can see that the same idea is being expressed, the "you" elided in the NASB/NIV where the idea is "How did it go with you, my daughter?"
A slightly more challenging word is the מִי. It is most often an interrogative statement of "who?" or "whoever?" HALOT notes it can mean "How?", but gives Ruth 3:16 as the only real example.1 This lack of support makes the rendering as "How?" as far less probable, even for Ruth 3:16. The NKJV, "Is that you?" carries the idea of "who is it," just less word for word in rendering.
But the translations using "how" are still communicating the idea behind the question, I believe, even if the question is most properly translated as noted "Who [are] you, my daughter?"
Naomi has not become blind, nor is it dark enough that she is unable to recognize her daughter-in-law. She is not saying: "Who are you? My daughter?" to find out information as to who just entered her house.
Rather, she is asking "Who are you?" in the sense of, "Who have you returned as from your mission I sent you on [cf. v.3-4]?" That is, are you now the woman of Boaz [espoused to Boaz] or not?
Ruth's answer was "Then she told her all that the man had done for her"(NKJV), which includes what he as to do to secure Ruth, v.12-13 (NKJV):
12 Now it is true that I am a close relative; however, there is a
relative closer than I. 13 Stay this night, and in the morning it
shall be that if he will perform the duty of a close relative for
you—good; let him do it. But if he does not want to perform the duty
for you, then I will perform the duty for you, as the LORD lives! Lie
down until morning.
So Ruth cannot answer Naomi's question directly as to who she is yet. She will either be the wife of Boaz, or become the wife of the closer kin.
Naomi's reply to Ruth shows that this information was divulged as part of the answer to Naomi's "Who are you?" question, for Naomi comforts Ruth in v.18 about not knowing who she is to be the wife of:
“Sit still, my daughter, until you know how the matter will turn out;
for the man will not rest until he has concluded the matter this day.”
That is, Naomi is saying by the end of this day, we both will know "Who you are to become the wife of" and thus "Who you will be."
The phrase also, from the stand point of the author of Ruth, provides a parallel to Boaz's question in 3:9, which is the same phrasing מִי־אָ֑תּ, "Who are you?" There, Boaz is indeed asking for Ruth's personal identity. Her reply is (NKJV):
“I am Ruth, your maidservant. Take your maidservant under your wing,
for you are a close relative.”
Ruth was seeking an identity change. Her reply sets up the contrast for the author's use of Naomi's question. When Naomi asks, she wants to know is Ruth still just Boaz's "maidservant" or has he indeed taken his maidservent under his wing? As Ruth was seeking an identity change, Naomi's is seeking through her question to know what her daughter-in-law's identity now is after her visit to Boaz.
Commentator's with Similar Observations
This idea has been recognized by others (all bold emphasis added by me). Paulus Cassel contributed to Lange's commentary for the book of Ruth, in which he notes:
When Ruth comes home, and Naomi asks, “Who art thou, my daughter,”
i.e. “how comest thou? as one whose claim has been acknowledged, or
That last phrase, "as one whose claim has been acknowledged, or otherwise?" shows he understands the question to be also about Ruth's status with respect to Boaz.
More recently, James Smith:
When Ruth approached the house, Naomi inquired, “Who are you, my
daughter?” The early morning darkness may have obscured the identity
of Ruth. More likely, however, Naomi’s question meant: Are you one
dishonored by rejection or one protected as a wife? She wanted to know
if the plan had worked!3
Marital status again is seen as the point of the question.
Warren Wiersbe summarizes the point likewise, and I like how he rephrased the question:
Naomi’s question in 3:16 has puzzled translators and interpreters. Why
would her own mother-in-law ask her who she was? The Living Bible
paraphrases the question, “Well, what happened, dear?” and both the
NIV and the NASB read, “How did it go, my daughter?” But the
Authorized Version translates the Hebrew text as it stands: “Who are
you, my daughter?” In other words, “Are you still Ruth the Moabitess,
or are you the prospective Mrs. Boaz?”4
The question is more focused than just a broad "How are you?" or "How did it go?" But rather, "Are you to be his wife or not?"
1 Ludwig Koehler, Walter Baumgartner, M. E. J. Richardson, and Johann Jakob Stamm, The Hebrew and Aramaic Lexicon of the Old Testament (Leiden: E.J. Brill, 1994–2000).
2 John Peter Lange, Philip Schaff, Paulus Cassel, and P. H. Steenstra, A Commentary on the Holy Scriptures: Ruth (1872; Bellingham, WA: Logos Bible Software, 2008).
3 James E. Smith, The Books of History, Old Testament Survey Series (Joplin, MO: College Press, 1995).
4 Warren W. Wiersbe, Be Committed, “Be” Commentary Series (Wheaton, IL: Victor Books, 1993).